Theoretical Background#

Lorenz-Mie Theory (LMT)#

Note

Lorenz-Mie Theory (LMT) provides the exact solution for the scattering of a plane wave by a spherical or cylindrical scatterer. The solution is expressed as an infinite summation, which is truncated in practical applications. PyMieSim solves these equations to retrieve several key scattering properties, such as scattering efficiencies and far-field intensities.

In PyMieSim, the spherical coordinates \(\theta\) and \(\phi\) are used to define the geometry, as depicted in the following figure:

https://github.com/MartinPdeS/PyMieSim/raw/master/docs/images/optical_setup.png

Key Relations Governing PyMieSim#

The following are some of the core relations used in PyMieSim to compute the scattered fields and related parameters.

\[ \begin{align}\begin{aligned}\begin{split}S_1 &= \sum\limits_{n=1}^{n_{\text{max}}} \frac{2n+1}{n(n+1)} \left( a_n \pi_n + b_n \tau_n \right) \\\end{split}\\\begin{split}S_2 &= \sum\limits_{n=1}^{n_{\text{max}}} \frac{2n+1}{n(n+1)} \left( a_n \tau_n + b_n \pi_n \right) \\\end{split}\\\begin{split}\text{Fields} &= E_{\theta}(\phi, \theta) \hat{\theta} + E_{\phi}(\phi, \theta) \hat{\phi} \\\end{split}\\\text{SPF} &= \sqrt{E_{\parallel}(\phi, \theta)^2 + E_{\perp}(\phi, \theta)^2}\end{aligned}\end{align} \]

Stokes Parameters:

These parameters describe the polarization state of the scattered light:

\[\begin{split}I &= |E_x|^2 + |E_y|^2 \\ Q &= |E_x|^2 - |E_y|^2 \\ U &= 2 \operatorname{Re}(E_x E_y^*) \\ V &= 2 \operatorname{Im}(E_x E_y^*)\end{split}\]

Scattering Properties#

Several important properties of the scatterer can be computed using PyMieSim. These include:

  • Scattering efficiency

  • Extinction efficiency

  • Absorption efficiency

  • Back-scattering efficiency

  • Forward-to-back scattering ratio

  • Radiation pressure efficiency

  • Anisotropy factor \(g\)

These properties are computed as follows:

\[\begin{split}Q_{\text{sca}} &= \frac{2}{x^2} \sum_{n=1}^{n_{\text{max}}} (2n+1)(|a_n|^2 + |b_n|^2) \\ Q_{\text{ext}} &= \frac{2}{x^2} \sum_{n=1}^{n_{\text{max}}} (2n+1) \operatorname{Re}(a_n + b_n) \\ Q_{\text{abs}} &= Q_{\text{ext}} - Q_{\text{sca}} \\ Q_{\text{back}} &= \frac{1}{x^2} \left| \sum_{n=1}^{n_{\text{max}}} (2n+1)(-1)^n (a_n - b_n) \right|^2 \\ Q_{\text{ratio}} &= \frac{Q_{\text{back}}}{Q_{\text{sca}}} \\ Q_{\text{pr}} &= Q_{\text{ext}} - g \cdot Q_{\text{sca}} \\ g &= \frac{4}{Q_{\text{sca}} x^2} \left[ \sum_{n=1}^{n_{\text{max}}} \frac{n(n+2)}{n+1} \operatorname{Re}(a_n a_{n+1}^* + b_n b_{n+1}^*) + \sum_{n=1}^{n_{\text{max}}} \frac{2n+1}{n(n+1)} \operatorname{Re}(a_n b_n^*) \right]\end{split}\]

The cross-sectional areas can be computed as:

\[\begin{split}A_s &= \pi r^2 \\ \sigma_i &= Q_i A_s\end{split}\]

Where: - \(C\) is the concentration of scatterers in the sample. - \(\sigma_{\text{sca}}\), \(\sigma_{\text{ext}}\), and \(\sigma_{\text{abs}}\) are the scattering, extinction, and absorption cross-sections, respectively.

An and Bn Coefficients#

The Lorenz-Mie theory also defines the coefficients \(a_n\) and \(b_n\), which characterize the scattered fields. These coefficients vary depending on the scatterer’s geometry.

### Sphere

For a spherical scatterer, the \(a_n\) and \(b_n\) coefficients are given by:

\[ \begin{align}\begin{aligned}\begin{split}a_n &= \frac{\mu_{\text{sp}} \Psi_n(x) \Psi_n'(Mx) - \mu M \Psi_n'(x) \Psi_n(Mx)}{\mu_{\text{sp}} \xi_n(x) \Psi_n'(Mx) - \mu M \xi_n'(x) \Psi_n(Mx)} \\\end{split}\\b_n &= \frac{\mu M \Psi_n(x) \Psi_n'(Mx) - \mu_{\text{sp}} \Psi_n'(x) \Psi_n(Mx)}{\mu M \xi_n(x) \Psi_n'(Mx) - \mu_{\text{sp}} \xi_n'(x) \Psi_n(Mx)}\end{aligned}\end{align} \]

With:

  • \(\psi_n\) and \(\xi_n\) being spherical Bessel and Hankel functions.

  • \(M = \frac{k_{\text{sp}}}{k}\) is the relative complex refractive index.

  • \(x = \frac{\pi d}{\lambda}\) is the size parameter.

  • \(\lambda\) is the wavelength in the surrounding medium.

Note: PyMieSim assumes \(\mu_{\text{sp}} = \mu\) for now, but this may change in future updates.

### Cylinder

For cylindrical scatterers, the coefficients are:

\[\begin{split}a_n &= \frac{M J_n(Mx) J_n'(mx) - m J_n'(Mx) J_n(mx)}{m J_n(Mx) H_n'(mx) - m J_n'(Mx) H_n(mx)} \\ b_n &= \frac{m J_n(Mx) J_n'(mx) - M J_n'(Mx) J_n(mx)}{M J_n(Mx) H_n'(mx) - m J_n'(Mx) H_n(mx)}\end{split}\]

Where: - \(M\) is the refractive index of the scatterer. - \(m\) is the refractive index of the medium. - \(H_n\) is the Hankel function of the first kind of order \(n\).

### Core/Shell Sphere

For core/shell spheres, the coefficients \(a_n\) and \(b_n\) are given by more complex expressions:

\[a_n &= \frac{\psi_n [\psi_n'(m_2 y) - A_n \chi_n'(m_2 y)] - m_2 \psi_n'(y) [\psi_n(m_2 y) - A_n \chi_n(m_2 y)]}{\xi_n(y) [\psi_n'(m_2 y) - A_n \chi_n'(m_2 y)] - m_2 \xi_n'(y) [\psi_n(m_2 y) - A_n \chi_n(m_2 y)]}\]

With: - \(x = \frac{2\pi R_{\text{core}}}{\lambda}\) and \(y = \frac{2\pi R_{\text{shell}}}{\lambda}\) - \(m_1 = \frac{n_{\text{core}}}{n_{\text{medium}}}\) and \(m_2 = \frac{n_{\text{shell}}}{n_{\text{medium}}}\)

Generalized Lorenz-Mie Theory (GLMT)#

Note

Coming soon — Future updates will cover the extension of LMT to more complex scatterers using GLMT.

Coupling Mechanisms#

There are two main coupling mechanisms for light collection: coherent coupling and non-coherent coupling.

Coherent coupling involves interference and usually results in less light being collected but offers additional information about the sample. Non-coherent coupling collects light without interference and is used by instruments like photodiodes.

The mathematical definitions are:

\[\begin{split}C_{\text{coh}} &= \left| \iint_{\Omega} \Phi_{\text{det}} \cdot \Psi_{\text{scat}}^* \, d\Omega \right|^2 \\ C_{\text{non-coh}} &= \iint_{\Omega} \left| \Phi_{\text{det}} \right|^2 \cdot \left| \Psi_{\text{scat}} \right|^2 \, d\Omega\end{split}\]

Centered Coupling assumes the scatterer is perfectly aligned with the detector. For non-centered scenarios, the footprint of the scatterer is defined by:

\[\eta_{l,m}(\delta_x, \delta_y) = \left| \mathcal{F}^{-1} \{ \Phi_{\text{det}} \cdot \Psi_{\text{scat}} \} \right|^2\]

The mean coupling can be calculated as:

\[\widetilde{\eta}_{l,m} = \langle \eta_{l,m} (\delta_x, \delta_y) \rangle\]